It's the battery - I've had exactly the same problem in the past. Batteries are OK when it's reasonably warm, but once it gets a bit parky, they give up the ghost very quickly. I used to have to jump start my bike in the winter mornings, but had no problems in the evening after it had spent all day lurking in an underground car park - a rise of a few degrees was all that was needed
Get an Optimate, or a new battery
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